I'm finding elements of your post confusing.
I have never heard of a "playlist folder". iTunes stores playlists in the main iTunes library.itl database file.
The easy way to get your phone to synch to your old library is to move the old one to the new computer:
iTunes: How to move [or copy] your music to a new computer [or another drive] - http://support.apple.com/kb/HT4527
Quick answer if you use iTunes default preferences settings: Copy the entire iTunes folder (and in doing so all its subfolders and files) intact to the other drive. Start iTunes with the option (shift on Windows) key held down and guide it to the new location of the library.
An idevice can only be synced to one computer at a time.
You can have up to 5 computers authorized for iTunes content at a time.
Heck - I confuse myself Thanks SO much for replying though!
Playlists - I copied the library.itl database file and my playlists did not move over. Should I try it again?
I just looked and in my Music folder I had Amazon MP3, iTunes, and Playlists. I was probably mistakenly thinking the Playlist folder was my iTunes playlists.
My music is all here, I just want my playlists
"An idevice can only be synced to one computer at a time" so once I synch with this new computer, I am fine?
Do you have a complete iTunes folder? One containing a library.itl file, Media folder, and an assortment of other things? iTunes is very fussy about everything being nice and neat and the best way to ensure that it it is happy is to move over a complete folder, not bits and pieces. That's why the instructions above say copy the whole iTunes folder.
I am guessing that "playlists" file/folder has something to do with Amazon or some other software you use.
What are the iTunes library files? - http://support.apple.com/kb/HT1660
More on iTunes library files and what they do - http://en.wikipedia.org/wiki/ITunes#Media_management
What are all those iTunes files? - http://www.macworld.com/article/139974/2009/04/itunes_files.html
iTunes 9 [and later]: Understanding iTunes Media Organization - http://support.apple.com/kb/ht3847
Image of iTunes folder structure Dec. 2011 - https://discussions.apple.com/message/16923545
Image of folder structure and explanation of different iTunes versions (turingtest2 post) - https://discussions.apple.com/message/13025536 and https://discussions.apple.com/message/17457605